Using the aerodynamic base bending moment or base torque as the input, the wind-induced response of a building can be computed using random vibration analysis by assuming idealized structural mode shapes, e.g. linear, and considering the special relationship between the aerodynamic moments and the generalized wind loads (e.g.,Tschanz and Davenport 1983; Zhou et al. 2001). However, instead of utilizing conventional generalized wind loads, a base-bending-moment-based procedure is suggested for evaluating equivalent static wind loads and response, which is computationally more efficient, especially in cases where mode shape correction is required (Zhou et al. 2001). Application of this procedure for the alongwind response has proven effective in recasting the traditional gust loading factor approach in a new format (Zhou et al. 1999; Zhou and Kareem 2001a). The procedure can be conveniently adapted to the acrosswind and torsional response (Zhou and Kareem 2001b).

Assuming a stationary Gaussian process, the expected maximum base bending moment response in the alongwind or acrosswind directions or the base torque response can be expressed in the following form:

(1)whereand= expected extreme value and mean of the moment or torque response, respectively,g= peak factor, andsM= root mean square (RMS) of the fluctuating base moment or base torque response, which can be computed bywhereSM(f)= power spectral density (PSD) of the fluctuating base moment or torque response. It can be shown that the PSD of the moment response can be computed by the following equation (Zhou and Kareem 2001a):

(2)where= structural transfer function of the first mode,f1andz1= natural frequency and critical damping ratio in the first mode, respectively, andSM(f)= PSD of the aerodynamic base moment or torque. The flexibility to consider non-ideal mode shapes and non-uniform mass distributions has been addressed in (2) (Boggs and Peterka 1989; Zhou et al. 1999, 2001). Note that the same symbol, but expressed in bold, is employed here to distinguish the base moment or base torque response from the external aerodynamic moment or torque. The former includes the dynamic amplification effects due to wind fluctuations and structural dynamics.

To facilitate computations, the integration in (2) is divided into two portions, i.e., background and resonant components. The resonant base moment or base torque response,, can be computed in closed-form by assuming that the excitation can be represented by white-noise in the vicinity of the structures natural frequency and by subsequently invoking the Residue Theorem for integration

(3)where= resonant peak factor,T= observation time, and subscriptR= resonant component. The background base moment and base torque,, is equal to the aerodynamic quantity since, for the background response, the structure responds statically with a dynamic magnification factor of unity and

(4)wheregB= background peak factor, which is usually at 3~4,sM= RMS aerodynamic moment, and subscriptB= background component. The mean base moment or base torque can be estimated from the HFBB test or by a mean pressure measurement test that is usually used as a companion test for the design of the cladding system.

Formulae for peak background and resonant moments using Aerodynamic Loads Database information

In this discussion and the accompanying online database, the measured aerodynamic loads are defined in terms of the RMS base moment coefficient and the non-dimensional Power Spectral Density (PSD) as follows:

(9)

(10)where= non-dimensional moment coefficient and= reference moment or torque, which is defined by,andfor the alongwind, acrosswind and torsional directions, respectively. Herer= air density,B= building width normal to the oncoming wind,D= building depth,= mean wind velocity evaluated at the building height,H, and subscriptsD,LandT= alongwind, acrosswind and torsional directions, respectively. It is noteworthy that, except for the alongwind load, different definitions have been used in the literature, e.g., theDis replaced with aBfor the acrosswind load, andBDis expressed in terms ofB2for the torque (Marukawa et al. 1992; Choi and Kanda 1993; Kijewski and Kareem 1998). These definitions will lead to significant differences in the RMS coefficients in (9), while having no impact on the normalized spectrum in (10).

The non-dimensional aerodynamic loads can be directly used in the above analysis procedure, e.g., the background and the resonant components of the base bending moment or base torque can be computed, respectively, by

(11)

(12)

Determination of equivalent static wind loads, accelerations and other response quantities

With the known base moment and base torque response, the equivalent static wind loading can be obtained by distributing the base moment to each floor in an appropriate manner as detailed in Zhou and Kareem (2001a, b). The equivalent static wind loads at height z for sway motions can be computed by

(5)while for the torsional case,

(6)where= resonant component of the equivalent static wind loading,m(z)= mass per unit height,I(z)= mass moment of inertia per unit height, andj1(z)= fundamental mode shape in the direction of motion.The wind-induced response, including the overall deflection, acceleration, internal forces, and stresses in each structural member can be computed using the equivalent static wind loads with a simple static analysis. For any response component, the resultant effect can be determined by summing the mean and the SRSS combination of the background and resonant components

(7)where= resultant wind-induced response of interest,,,= mean, peak background and peak resonant response components, which are computed using the corresponding equivalent static wind load component with static analysis.However, for the acceleration response, only the resonant component is of interest. For example, the peak acceleration in the sway mode can be computed by

(8)In the case of the torsional response, the mass per unit height,m(z), in (8) is replaced byI(z), the mass moment of inertia per unit height. The resulting RMS acceleration can then be determined by dividing the peak accelerations by the resonant peak factor,gR.